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4.8E: Exercises for Section 4.8

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    In exercises 1 - 6, evaluate the limit.

    1) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{e^x}{x}\).

    2) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k}\).

    Answer
    \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k} \quad = \quad ∞\)

    3) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{\ln x}{x^k}\).

    4) Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}\).

    Answer
    \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}\)

    5. Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^3−a^3}\).

    6. Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n}\).

    Answer
    \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}\)

    In exercises 7 - 11, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.

    7) \(\displaystyle \lim_{x→0^+}x^2\ln x\)

    8) \(\displaystyle \lim_{x→∞}x^{1/x}\)

    Answer
    Cannot apply directly; use logarithms

    9) \(\displaystyle \lim_{x→0}x^{2/x}\)

    10) \(\displaystyle \lim_{x→0}\frac{x^2}{1/x}\)

    Answer
    Cannot apply directly; rewrite as \(\displaystyle \lim_{x→0}x^3\)

    11) \(\displaystyle \lim_{x→∞}\frac{e^x}{x}\)

    In exercises 12 - 40, evaluate the limits with either L’Hôpital’s rule or previously learned methods.

    12) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}\)

    Answer
    \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3} \quad = \quad 6\)

    13) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}\)

    14) \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}\)

    Answer
    \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2\)

    15) \(\displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}\)

    16) \(\displaystyle \lim_{x→π}\frac{x−π}{\sin x}\)

    Answer
    \(\displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1\)

    17) \(\displaystyle \lim_{x→1}\frac{x−1}{\sin x}\)

    18) \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}\)

    Answer
    \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n\)

    19) \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}\)

    20) \(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}\)

    Answer
    \(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}\)

    21) \(\displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}\)

    22) \(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}\)

    Answer
    \(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}\)

    23) \(\displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}\)

    24) \(\displaystyle \lim_{x→1}\frac{x-1}{\ln x}\)

    Answer
    \(\displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1\)

    25) \(\displaystyle \lim_{x→0}\,(x+1)^{1/x}\)

    26) \(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}\)

    Answer
    \(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}\)

    27) \(\displaystyle \lim_{x→0^+}x^{2x}\)

    28) \(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)\)

    Answer
    \(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1\)

    29) \(\displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}\)

    30) \(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right)\)

    Answer
    \(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0\)

    31) \(\displaystyle \lim_{x→∞}(x−e^x)\)

    32) \(\displaystyle \lim_{x→∞}x^2e^{−x}\)

    Answer
    \(\displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0\)

    33) \(\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}\)

    34) \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}\)

    Answer
    \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1\)

    35) \(\displaystyle \lim_{x→π/4}(1−\tan x)\cot x\)

    36) \(\displaystyle \lim_{x→∞}xe^{1/x}\)

    Answer
    \(\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞\)

    37) \(\displaystyle \lim_{x→0}x^{1/\cos x}\)

    38) \(\displaystyle \lim_{x→0^{+} }x^{1/x}\)

    Answer
    \(\displaystyle \lim_{x→0^{+} }x^{1/x} \quad = \quad 0\)

    39) \(\displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x\)

    40) \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x\)

    Answer
    \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}\)

    For exercises 41 - 50, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.

    41) [T] \(\displaystyle \lim_{x→0}\frac{e^x−1}{x}\)

    42) [T] \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)\)

    Answer
    \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0\)

    43) [T] \(\displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}\)

    44) [T] \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}\)

    Answer
    \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1\)

    45) [T] \(\displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}\)

    46) [T] \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}\)

    Answer
    \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0\)

    47) [T] \(\displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)\)

    48) [T] \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right)\)

    Answer
    \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1\)

    49) [T] \(\displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}\)

    50) [T] \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}\)

    Answer
    \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2\)

    4.8E: Exercises for Section 4.8 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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